Speech analysis involves obtaining characteristics of a speech signal for use in speech-enabled applications, such as speech synthesis, speech recognition, speaker verification and identification, and enhancement of speech signal quality. Speech analysis is particularly important to speech coding systems.
Speech coding refers to the techniques and methodologies for efficient digital representation of speech and is generally divided into two types, waveform coding systems and model-based coding systems. Waveform coding systems are concerned with preserving the waveform of the original speech signal. One example of a waveform coding systems is the direct sampling system which directly samples a sound at high bit rates (“direct sampling systems”). Direct sampling systems are typically preferred when quality reproduction is especially important. However, direct sampling systems require a large bandwidth and memory capacity. A more efficient example of waveform coding is pulse code modulation.
In contrast, model-based speech coding systems are concerned with analyzing and representing the speech signal as the output of a model for speech production. This model is generally parametric and includes parameters that preserve the perceptual qualities and not necessarily the waveform of the speech signal. Known model-based speech coding systems use a mathematical model of the human speech production mechanism referred to as the source-filter model.
The source-filter model models a speech signal as the air flow generated from the lungs (an “excitation signal”), filtered with the resonances in the cavities of the vocal tract, such as the glottis, mouth, tongue, nasal cavities and lips (a “filter”). The excitation signal acts as an input signal to the filter similarly to the way the lungs produce air flow to the vocal tract. Model-based speech coding systems using the source-filter model, generally determine and code the parameters of the source-filter model. These model parameters generally include the parameters of the filter. The model parameters are determined for successive short time intervals or frames (e.g., 10 to 30 ms analysis frames), during which the model parameters are assumed to remain fixed or unchanged. However, it is also assumed that the parameters will change with each successive time interval to produce varying sounds.
The parameters of the model are generally determined through analysis of the original speech signal. Because the filter (the “analysis filter”) generally includes a polynomial equation including several coefficients to represent the various shapes of the vocal tract, determining the parameters of the filter generally includes determining the coefficients of the polynomial equation (the “filter coefficients”). Once the filter coefficients have been obtained, the excitation signal can be determined by filtering the original speech signal with a second filter that is the inverse of the filter.
One method for determining the coefficients of the filter is through the use of linear predictive analysis (“LPA”) techniques. LPA is a time-domain technique based on the concept that during a successive short time interval or frame “N,” each sample of a speech signal (“speech signal sample” or “s[n]”) is predictable through a linear combination of samples from the past s[n−k] together with the excitation signal u[n].
                              s          ⁡                      [            n            ]                          =                                            ∑                              k                =                1                            M                        ⁢                                                  ⁢                                          a                k                            ⁢                              s                ⁡                                  [                                      n                    -                    k                                    ]                                                              +                      G            ⁢                                                  ⁢                          u              ⁡                              [                n                ]                                                                        (        1        )            where G is a gain term representing the loudness over the frame (about 10 ms), M is the order of the polynomial (the “prediction order”), and ak are the filter coefficients which are also referred to as the “LP coefficients.” The analysis filter is therefore a function of the past speech samples s[n] and is represented in the z-domain by the formula:H[z]=G/A[z]  (2)A[z] is an M order polynomial given by:
                              A          ⁡                      [            z            ]                          =                  1          +                                    ∑                              k                =                1                            M                        ⁢                                                  ⁢                                          a                k                            ⁢                              z                                  -                  k                                                                                        (        3        )            The order of the polynomial A[z] can vary depending on the particular application, but a 10th order polynomial is commonly used with an 8 kHz sampling rate.
The LP coefficients a1 . . . aM are computed by analyzing the actual speech signal s[n]. The LP coefficients are approximated as the coefficients of a filter used to reproduce s[n] (the “synthesis filter”). The synthesis filter uses the same LP coefficients as the analysis filter and produces a synthesized version of the speech signal. The synthesized version of the speech signal may be estimated by a predicted value of the speech signal {tilde over (s)}[n]. {tilde over (s)}[n] is defined according to the formula:
                                          s            ~                    ⁡                      [            n            ]                          =                  -                                    ∑                              k                =                1                            M                        ⁢                                                  ⁢                                          a                k                            ⁢                              s                ⁡                                  [                                      n                    -                    k                                    ]                                                                                        (        4        )            
Because s[n] and s[n] are not exactly the same, there will be an error associated with the predicted speech signal s[n] for each sample n referred to as the prediction error ep[n], which is defined by the equation:
                                          e            p                    ⁡                      [            n            ]                          =                                            s              ⁡                              [                n                ]                                      -                                          s                ~                            ⁡                              [                n                ]                                              =                                    s              ⁡                              [                n                ]                                      +                                          ∑                                  k                  =                  1                                M                            ⁢                                                          ⁢                                                a                  k                                ⁢                                  s                  ⁡                                      [                                          n                      -                      k                                        ]                                                                                                          (        5        )            where the sum of all the prediction errors defines the total prediction error Ep:Ep=Σep2[k]  (6)where the sum is taken over the entire speech signal. The LP coefficients a1 . . . aM are generally determined so that the total prediction error Ep is minimized (the “optimum LP coefficients”).
One common method for determining the optimum LP coefficients is the autocorrelation method. The basic procedure consists of signal windowing, autocorrelation calculation, and solving the normal equation leading to the optimum LP coefficients. Windowing consists of breaking down the speech signal into frames or intervals that are sufficiently small so that it is reasonable to assume that the optimum LP coefficients will remain constant throughout each frame. During analysis, the optimum LP coefficients are determined for each frame. These frames are known as the analysis intervals. The LP coefficients obtained through analysis are then used for synthesis or prediction inside frames known as synthesis intervals. In practice, the analysis and synthesis intervals might not be the same.
When windowing is used, assuming for simplicity a rectangular window sequence of unity height including window samples w[n], the total prediction error Ep in a given frame or interval may be expressed as:
                              E          p                =                              ∑                          k              =                              n                1                                                    n              2                                ⁢                                          ⁢                                    e              p              2                        ⁡                          [              k              ]                                                          (        7        )            where n1 and n2 are the indexes corresponding to the beginning and ending samples of the window sequence and define the synthesis frame.
Once the speech signal samples s[n] are isolated into frames, the optimum LP coefficients can be found using an autocorrelation method. To minimize the total prediction error, the values chosen for the LP coefficients must cause the derivative of the total prediction error with respect to each LP coefficients to equal or approach zero. Therefore, the partial derivative of the total prediction error is taken with respect to each of the LP coefficients, producing a set of M equations. Fortunately, these equations can be used to relate the minimum total prediction error to an autocorrelation function:
                                          E            p                    =                                                    R                p                            ⁡                              [                0                ]                                      -                                          ∑                                  k                  =                  1                                M                            ⁢                                                          ⁢                                                a                  i                                ⁢                                  R                                      p                    [                                                  ⁢                k                                                    ]                            (        8        )            where M is the prediction order and Rp(k) is an autocorrelation function for a given time-lag l which is expressed by:
                              R          ⁡                      [            l            ]                          =                              ∑                          k              =              1                                      N              -              1                                ⁢                                    w              ⁡                              [                k                ]                                      ⁢                          s              ⁡                              [                k                ]                                      ⁢                          w              ⁡                              [                                  k                  -                  l                                ]                                      ⁢                          s              ⁡                              [                                  k                  -                  l                                ]                                                                        (        9        )            where s[k] are speech signal sample, w[k] are the window samples that together form a plurality of window sequences each of length N (in number of samples) and s[k−l] and w[k−l] are the input signal samples and the window samples lagged by l. It is assumed that w[n] may be greater than zero only from k=0 to N−1.
Because the minimum total prediction error can be expressed as an equation in the form Ra=b (assuming that Rp[0] is separately calculated), the Levinson-Durbin algorithm may be used to determine for the optimum LP coefficients.
Many factors affect the minimum total prediction error that can be achieved including the shape of the window in the time domain. Generally, the window sequences adopted by coding standards have a shape that includes tapered-ends so that the amplitudes are low at the beginning and end of the window sequences with a peak amplitude located in-between. These windows are described by simple formulas and their selection inspired by the application in which they will be used. Generally, known methods for choosing the shape of the window are heuristic. There is no deterministic method for determining the optimum window shape.